The discussion of Wittgenstein's account of quantification in Chapter 20 left unaddressed what sort of theory of types it commits us to. To answer this question we need to look in more detail at the ...
More

The discussion of Wittgenstein's account of quantification in Chapter 20 left unaddressed what sort of theory of types it commits us to. To answer this question we need to look in more detail at the motivation for believing in logical types at all. That motivation derives from Russell's paradox, the problem which had originally attracted Wittgenstein's notice back in 1909. This chapter discusses Russell's theory of types, Wittgenstein's vicious circle principle, types as classes of propositions, types and molecular propositions, types and generality, uniting generality and truth-functions, the general form of proposition, and unsayability.Less

Resolving the paradoxes

Michael Potter

Published in print: 2008-10-01

The discussion of Wittgenstein's account of quantification in Chapter 20 left unaddressed what sort of theory of types it commits us to. To answer this question we need to look in more detail at the motivation for believing in logical types at all. That motivation derives from Russell's paradox, the problem which had originally attracted Wittgenstein's notice back in 1909. This chapter discusses Russell's theory of types, Wittgenstein's vicious circle principle, types as classes of propositions, types and molecular propositions, types and generality, uniting generality and truth-functions, the general form of proposition, and unsayability.

Here the concern is with the logic underlying Whitehead and Russell's Principia Mathematica and with the relation of that logic to Russell's underlying metaphysics. The author emphasizes the fact ...
More

Here the concern is with the logic underlying Whitehead and Russell's Principia Mathematica and with the relation of that logic to Russell's underlying metaphysics. The author emphasizes the fact that work is, strictly speaking, a theory of propositional functions, not of classes; sentences containing symbols for classes are defined by means of propositional functions. It is terms of the latter sort of entity that Russell's Paradox must be solved; the theory of types is, strictly speaking, a theory of the stratification of propositional functions.Less

The Logic of Principia Mathematica

Peter Hylton

Published in print: 1992-11-26

Here the concern is with the logic underlying Whitehead and Russell's Principia Mathematica and with the relation of that logic to Russell's underlying metaphysics. The author emphasizes the fact that work is, strictly speaking, a theory of propositional functions, not of classes; sentences containing symbols for classes are defined by means of propositional functions. It is terms of the latter sort of entity that Russell's Paradox must be solved; the theory of types is, strictly speaking, a theory of the stratification of propositional functions.

Merely taking the symbolic turn — to conceive of a proposition as symbolizing what it expresses, rather than being identical with it — is not yet to go very far towards uncovering the structure of ...
More

Merely taking the symbolic turn — to conceive of a proposition as symbolizing what it expresses, rather than being identical with it — is not yet to go very far towards uncovering the structure of propositions. In his letters to Russell during 1912, Wittgenstein was still operating with a Russellian conception of that structure. A proposition he still thought of as a sort of complex. This chapter discusses Wittgenstein's rejection of the idea that propositions consist of names related by a copula, citing that there cannot be two different types of things and that the theory of types is superfluous.Less

Unity

Michael Potter

Published in print: 2008-10-01

Merely taking the symbolic turn — to conceive of a proposition as symbolizing what it expresses, rather than being identical with it — is not yet to go very far towards uncovering the structure of propositions. In his letters to Russell during 1912, Wittgenstein was still operating with a Russellian conception of that structure. A proposition he still thought of as a sort of complex. This chapter discusses Wittgenstein's rejection of the idea that propositions consist of names related by a copula, citing that there cannot be two different types of things and that the theory of types is superfluous.

The organizing principles of the simple theory of types are: (i) sentence-meanings are truth-values; (ii) the meaning of a name is its referent; and (iii) the meanings of other basic constituents of ...
More

The organizing principles of the simple theory of types are: (i) sentence-meanings are truth-values; (ii) the meaning of a name is its referent; and (iii) the meanings of other basic constituents of sentences are to be assigned in such a way that when functions are applied to arguments as the structure of the sentence dictates, the final output is a truth-value. This chapter discusses extensional type-theory covering intransitive verbs, transitive verbs, binary truth-functions, common nouns and adjectives, quantified noun phrases, the lambda operator, and type shifting and systematic ambiguity. It also discusses hyperintensional semantics.Less

A Brief Guide to Type Theory

Graeme Forbes

Published in print: 2006-06-29

The organizing principles of the simple theory of types are: (i) sentence-meanings are truth-values; (ii) the meaning of a name is its referent; and (iii) the meanings of other basic constituents of sentences are to be assigned in such a way that when functions are applied to arguments as the structure of the sentence dictates, the final output is a truth-value. This chapter discusses extensional type-theory covering intransitive verbs, transitive verbs, binary truth-functions, common nouns and adjectives, quantified noun phrases, the lambda operator, and type shifting and systematic ambiguity. It also discusses hyperintensional semantics.

In this chapter the quotients of a given theory of presheaf type are investigated by means of Grothendieck topologies that can be naturally attached to them, establishing a ‘semantic’ representation ...
More

In this chapter the quotients of a given theory of presheaf type are investigated by means of Grothendieck topologies that can be naturally attached to them, establishing a ‘semantic’ representation for the classifying topos of such a quotient as a subtopos of the classifying topos of the given theory of presheaf type. It is also shown that the models of such a quotient can be characterized among the models of the theory of presheaf type as those which satisfy a key property of homogeneity with respect to a Grothendieck topology associated with the quotient. A number of sufficient conditions for the quotient of a theory of presheaf type to be again of presheaf type are also identified: these include a finality property of the category of models of the quotient with respect to the category of models of the theory and a rigidity property of the Grothendieck topology associated with the quotient.Less

Quotients of a theory of presheaf type

Olivia Caramello

Published in print: 2017-12-21

In this chapter the quotients of a given theory of presheaf type are investigated by means of Grothendieck topologies that can be naturally attached to them, establishing a ‘semantic’ representation for the classifying topos of such a quotient as a subtopos of the classifying topos of the given theory of presheaf type. It is also shown that the models of such a quotient can be characterized among the models of the theory of presheaf type as those which satisfy a key property of homogeneity with respect to a Grothendieck topology associated with the quotient. A number of sufficient conditions for the quotient of a theory of presheaf type to be again of presheaf type are also identified: these include a finality property of the category of models of the quotient with respect to the category of models of the theory and a rigidity property of the Grothendieck topology associated with the quotient.

This chapter carries out a systematic investigation of the class of geometric theories of presheaf type (i.e. classified by a presheaf topos), by using in particular the results on flat functors ...
More

This chapter carries out a systematic investigation of the class of geometric theories of presheaf type (i.e. classified by a presheaf topos), by using in particular the results on flat functors established in Chapter 5. First, it establishes a number of general results on theories of presheaf type, notably including a definability theorem and a characterization of the finitely presentable models of such a theory in terms of formulas satisfying a key property of irreducibility. Then it presents a fully constructive characterization theorem providing necessary and sufficient conditions for a theory to be of presheaf type expressed in terms of the models of the theory in arbitrary Grothendieck toposes. This theorem is shown to admit a number of simpler corollaries which can be effectively applied in practice for testing whether a given theory is of presheaf type as well as for generating new examples of such theories.Less

Theories of presheaf type: general criteria

Olivia Caramello

Published in print: 2017-12-21

This chapter carries out a systematic investigation of the class of geometric theories of presheaf type (i.e. classified by a presheaf topos), by using in particular the results on flat functors established in Chapter 5. First, it establishes a number of general results on theories of presheaf type, notably including a definability theorem and a characterization of the finitely presentable models of such a theory in terms of formulas satisfying a key property of irreducibility. Then it presents a fully constructive characterization theorem providing necessary and sufficient conditions for a theory to be of presheaf type expressed in terms of the models of the theory in arbitrary Grothendieck toposes. This theorem is shown to admit a number of simpler corollaries which can be effectively applied in practice for testing whether a given theory is of presheaf type as well as for generating new examples of such theories.

This chapter discusses several classical as well as new examples of theories of presheaf type from the perspective of the theory developed in the previous chapters. The known examples of theories of ...
More

This chapter discusses several classical as well as new examples of theories of presheaf type from the perspective of the theory developed in the previous chapters. The known examples of theories of presheaf type that are revisited in the course of the chapter include the theory of intervals (classified by the topos of simplicial sets), the theory of linear orders, the theory of Diers fields, the theory of abstract circles (classified by the topos of cyclic sets) and the geometric theory of finite sets. The new examples include the theory of algebraic (or separable) extensions of a given field, the theory of locally finite groups, the theory of vector spaces with linear independence predicates and the theory of lattice-ordered abelian groups with strong unit.Less

Examples of theories of presheaf type

Olivia Caramello

Published in print: 2017-12-21

This chapter discusses several classical as well as new examples of theories of presheaf type from the perspective of the theory developed in the previous chapters. The known examples of theories of presheaf type that are revisited in the course of the chapter include the theory of intervals (classified by the topos of simplicial sets), the theory of linear orders, the theory of Diers fields, the theory of abstract circles (classified by the topos of cyclic sets) and the geometric theory of finite sets. The new examples include the theory of algebraic (or separable) extensions of a given field, the theory of locally finite groups, the theory of vector spaces with linear independence predicates and the theory of lattice-ordered abelian groups with strong unit.

Wittgenstein’s account of propositional representation imposes limits on what propositions can represent. This chapter explores Wittgenstein’s applications of this result. It considers first ...
More

Wittgenstein’s account of propositional representation imposes limits on what propositions can represent. This chapter explores Wittgenstein’s applications of this result. It considers first Wittgenstein’s treatment of Russell’s paradox and the theory of types. Then it looks at his discussion of internal properties, relations, and concepts, concentrating on the claim that there cannot be propositions representing their instantiation. The chapter considers next whether the combinatorial families under which Wittgenstein would place the constituents of facts and propositions can be expected to correspond to the traditional ontological categories (individual, property, binary relation…). It looks next at Wittgenstein’s contention that illogical thought is impossible. The chapter returns then to the substance passage, arguing that the claim that the world has substance has to be understood as ascribing to the possibilities of combinations of objects into states of affairs, and the status of internal properties and relations.Less

The Limits of Representation

José L. Zalabardo

Published in print: 2015-08-01

Wittgenstein’s account of propositional representation imposes limits on what propositions can represent. This chapter explores Wittgenstein’s applications of this result. It considers first Wittgenstein’s treatment of Russell’s paradox and the theory of types. Then it looks at his discussion of internal properties, relations, and concepts, concentrating on the claim that there cannot be propositions representing their instantiation. The chapter considers next whether the combinatorial families under which Wittgenstein would place the constituents of facts and propositions can be expected to correspond to the traditional ontological categories (individual, property, binary relation…). It looks next at Wittgenstein’s contention that illogical thought is impossible. The chapter returns then to the substance passage, arguing that the claim that the world has substance has to be understood as ascribing to the possibilities of combinations of objects into states of affairs, and the status of internal properties and relations.